User blog:Primussupremus/Array notation from my website endless possiblities.
This is (part of) the array of the Array notation I have started to define on my site endless possibilities. {a} = a. {a,b} = a^b {a,b,c} = a↑(b c = a{b up arrow's}c {a,b,c,d} = {a,b,c} recursed d-1 times for d>1. for example: {3,3,3,3} = 3↑(3↑(3↑(3)3)3)3. In this way we can reach numbers like grahams number of {3,4,3,64} with ease. After thinking for a bit back when I first started developing this notation I decided to change things was I got to a 5 entry array and above. {a,b,c,d,e} = {a,b,c,d} recursed ({a,b,c,d}-1) times for e-1 repeats. With that I then defined k tuples or arrays of length>=5 to be: {k-1 tuple} recursed ({k-1 tuple}-1) times for the limit (entry/symbol)-1 repeats. As shown above: with {a,b,c,d,e}. Next I introduced a symbol & to represent linear arrays of any size. {K&L} = an array of length k made out of some symbol L for example: {2&5} = {5,5} = 5^5 = 3125. (The next parts of my notation emphasized the usage of repeating something n-1 times.) Beyond K&L I introduced a new symbol into my notation P. {(K&L)&P} = {K&L} recursed ({K&L}-1) times for P-1 repeats. Next I introduced a symbol Q to increase the strength of the notation even further: {((K&L)&P)&Q} = {(K&L)&P} recursed ({(K&L)&P}-1) times for Q-1 repeats. I then generalized arrays of this sort as: "An array closed under the & sign as shown above of length K>2 = the (k-1)th array of this format (where k is the length of the array you are using) recursed (k-1)th array , number of times for X-1 repeats (where X is the symbol outside of the brackets and connected to whats inside the brackets, for example {(K&L)P} includes such a symbol called P." I then demonstrated a way to compact arrays of this sort into digestible chunks (as one might say: {K(X)K} = an array of length X made out of some symbol K , where the array is of the kind shown above. for example: {2(2)2} = {2&2} = {2,2} = 2^2 = 4. This kind of array is much stronger than the previously defined array's but we can go further by making {K(X)K} the input of X. {K({K(X)K})K} = {K(X)K} recursed ({K(X)K}-1) times. {K({K({K(X)K})K})K}= {K({K(X)K})K} recursed ({K({K(X)K})K}-1) times. I called these types of arrays wingspan array for the obvious reason that they look like wings. the first type: {K(X)K} I called an array of wingspan 1 the second type: {K({K(X)K})K} an array of wingspan 2 the third type: {K({K({K(X)K})K})K} an array of wingspan 3. I then made up a new type of array to generalize wingspan arrays: {K++:(X)} where X refers to the wingspan of the array and K is the symbol to be used in said array. for example: {2++:(1)} = {2(2)2} = {2&2} = {2,2} = 4. I then defined things like: {K({K++:(X)})K} = {K++:(X)} recursed ({K++:(X)}-1) times. Of course you'll probably understand what's coming next as: {K({K({K++:(X)})K})K} = {K({K++:(X)})K} recursed ({K({K++:(X)})K}-1) times. These types of arrays are similar to wingspan arrays but far stronger so I called them meta wing span arrays. The previous two examples were meta wing span arrays of lengths 1 and 2. Next I defined a way to generalize wingspan arrays by having: {K**:(X)} where X refers to the wingspan of the meta-array and K refers to the symbol required for that array. After that I incremented {K**:(X)} to reach things like: {K({K**:(X)})K} = {K**:(X)} recursed ({K**:(X)}-1) times. I then decided to call arrays of this sort super wingspan arrays. I introduced the symbol ↑(1) to generalize super arrays. {K:↑(1)(X)} = a super wingspan array of length X made out of some symbol K. Next:{K:↑(2)(X)} = {K:↑(1)(X)} recursed ({K:↑(1)(X)}-1) times. Since I've defined what ↑(1) and ↑(2) does to my notation I can then define what ↑(N) does. {K:↑(N)(X)} = {K:↑(N-1)(X)} recursed ({K:↑(N-1)(X)}-1) times. This sort of array where the up-arrow has an entry value of N can be written as: {K:↑(ω)(X)} = {K:↑(N)(X)}. (where N = X) As we've reached the first ordinal part of my notation where we use omega I think that I'll stop there just for now. Category:Blog posts